Geometrical properties of optimal Volterra filters for signal detection

Linear-quadratic filters are a special example of Volterra filters that are limited to the second order. It is shown that all the results recently published which are valid in the linear-quadratic case can be extended with the appropriate notations to Volterra filters of arbitrary order. Particularly, the optimum Volterra filter giving the maximum of the deflection for detecting a signal in noise is wholly calculated. In addition, several geometrical properties of optimal Volterra filters are investigated by introducing appropriate scalar products. In particular, the concept of space orthogonal to the signal and the noise alone reference (NAR) property are introduced, allowing a decomposition of the optimal filter that exhibits a relation between detection and estimation. Extensions to the infinite case and relations with the likelihood ratio are also investigated